J. Nig. Soc. Phys. Sci. 2 (2020) 26–35

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

On Fuzzy Soft Set-Valued Maps with Application

Mohammed Shehu Shagari∗

Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Nigeria

Abstract

Soft set and fuzzy soft set theories are proposed as mathematical tools for dealing with uncertainties. There has been tremendous progress con-
cerning the extensions of these theories from different point of views of researchers so as to accommodate more robust and expressive applications
in everyday life. In line with this development, in this paper, we combine the two aforementioned notions to initiate a novel concept of set-valued
maps whose range set is a family of fuzzy soft sets. The later idea is employed to define Suzuki-type fuzzy soft (e, K)-weak contractions, thereby
establishing some related fuzzy soft fixed point theorems. As a consequence, several well-known Suzuki-type fixed point theorems are derived
as corollaries. Examples are also provided to validate the new concepts and to support the authenticity of the obtained results. Moreover, an
application in homotopy is considered to show the usability of the obtained results.

Keywords: Soft set, Fuzzy soft set, Fuzzy soft set-valued map, Fuzzy soft fixed point, Weak contraction

Article History :
Received: 30 January 2020
Received in revised form: 14 February 2020
Accepted for publication: 15 February 2020
Published: 28 February 2020

c©2020 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: B. J. Falaye

1. Introduction and preliminaries

The classical Banach contraction theorem [1] is one of the
extremely useful results in nonlinear functional analysis. In the
setting of a metric space , this theorem is stated as follows.

Theorem 1.1. [1] Let (X, d) be a complete metric space and
T : X −→ X be a contraction, i.e., there exists an α ∈ (0, 1)
such that

d(T x, T y) ≤ αd(x, y), for all x, y ∈ X. (1)

Then

(i) T has a unique fixed point u in X;

∗Corresponding author tel. no:
Email address: ssmohammed@abu.edu.ng (Mohammed Shehu Shagari)

(ii) The Picard iteration {xn}∞n=0 , given by

xn+1 = T xn, n = 0, 1, 2, · · ·

converges to u, for any x0 in X.

Theorem 1.1 has several generalizations, see, for example,[2,
3, 4, 5, 6, 7] and the references therein. The two well-known
drawbacks of Theorem 1.1 are:

(drb1) The contractive condition in (1) compels T to be contin-
uous;

(drb2) The theorem cannot characterize metric completeness of
X.

Problems (drb1)-(drb2) were resolved affirmatively by Kannan
[18]. Recall that a mapping T (not necessarily continuous) on a
metric space X is said to be a Kannan contraction if there exists
an α ∈

[
0, 12

)
such that

d(T x, T y) ≤ α
[
d(x, T x) + d(y, T y)

]
, for all x, y ∈ X. (2)

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M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 27

This achievement in Kannan contraction was first noted by Sub-
rahmanyam [28] that a metric space X is complete if and only
if every Kannan contraction on X has a fixed point. Following
(2), more than a handful of papers were devoted to studying
fixed point theorems for classes of contractive type conditions
that do not require the continuity of T ; see, for instance,[8, 9,
10, 11, 12]. However, researchers noticed that Kannan’s the-
orem is not a generalization of Theorem 1.1. Along the way,
the notion of weak contraction was introduced by Berinde [13].
The idea generalized the well-celebrated fixed point theorems
due to Banach [1], Chatterjea [9], Zamfirescu [12], and many
others.

Definition 1.2. Let (X, d) be a metric space. A mapping T :
X −→ X is called weak contraction if there exists a constants
α ∈ (0, 1) and K ≥ 0 such that

d(T x, T y) ≤ αd(x, y) + Kd(y, T x), for all x, y ∈ X. (3)

Thereafter, in 2007, M.Berinde and V. Berinde [14] extended
the concept of weak contraction from the case of single-valued
mappings to multi-valued mappings and established some con-
vergence theorems for the Picard iteration in connection with
multi-valued weak contraction.

Definition 1.3. Let (X, d) be a metric space and T : X −→
C B(X) be a multi-valued mapping. Then T is called a multi-
valued weak contraction or multi-valued (θ, L)-weak contrac-
tion if and only if there exist constants θ ∈ (0, 1) and K ≥ 0
such that for all x, y ∈ X,

H(T x, T y) ≤ θd(x, y) + Ld(y, T x). (4)

Furthermore, a notable attempt at resolving problems (drb1)-
(drb2) was presented in 2008 by Suzuki [29]. The following is
known in the literature as Suzuki fixed point theorem.

Theorem 1.4. Let (X, d) be a complete metric space and T be
a mapping on X. Define a nondecreasing function r from [0, 1)
onto

(
1
2 , 1

]
by

r(t) =


1, if 0 ≤ t ≤

√
5−1
2

(1−t)
r2 , if

√
5−1
2 ≤ r ≤

1
√

2
1

(1+t) , if
1
√

2
≤ r < 1.

Assume that there exists t ∈ [0, 1) such that

r(t)d(x, T x) ≤ d(x, y)

implies

d(T x, T y) ≤ rd(x, y), for all x, y ∈ X.

Then T has a unique fixed point u in X. Moreover, T n x −→ u
as n −→∞ for all x ∈ X.

An interesting improvement of Suzuki fixed point theorem
in the setting of multivalued mappings is due to Djoric and La-
zovic [16]. In subsequent section, we shall derive the main re-
sult in [16] as a consequence of our result.

The real world is filled with uncertainty, vagueness and im-
precision. The notions we meet in everyday life are vague rather
than precise. In recent time, researchers have taken keen inter-
ests in modelling vagueness due to the fact that many practi-
cal problems within fields such as biology, economics, engi-
neering, environmental sciences, medical sciences involve data
containing various forms of uncertainties. To handle these com-
plications, one cannot successfully employ classical mathemat-
ical methods due to the presence of different kinds of incom-
plete knowledge typical of these mix-ups. Earlier in the liter-
ature, there are four known theories for dealing with imperfect
knowledge, namely Probability Theory (PT), Fuzzy Set The-
ory (FST)[30], Interval Mathematics and Rough Set Theory
(RST)[27]. All the aforementioned tools require pre-assignment
of some parameters; for example, membership function in FST,
probability density function in PT and equivalent relation in
RST. Such pre-specifications, viewed in the backdrop of incom-
plete knowledge, give rise to every day problems. In this con-
cern, Molodstov [26] initiated the concept of Soft Set Theory
(SST) with the aim of handling phenomena and notions of am-
biguous, undefined and imprecise environments. Hence, SST
does not need the pre-specifications of a parameter, rather it
accommodates approximate descriptions of objects. In other
words, one can use any suitable parametrization tool with the
help of words, sentences, real numbers, mappings, and so on.
Thereby, making SST an adequate formalism for approximate
reasoning. In the pioneer work of Molodstov [26], several po-
tential applications of SST have been pointed out in the ar-
eas of Riemann integration, smoothness of functions, theory of
probability, theory of measurement, game theory and operation
research . Interestingly, Molodstov [26] emphasized that the
models by fuzzy sets and soft sets are interrelated. Yang et al
[22] emphasized that SST needed to be expanded in different
directions to extend its applications to other fields. Along the
lane, by combining the ideas of soft sets and fuzzy sets, Maji et
al [21] initiated the concept of fuzzy soft sets and discussed its
various properties. Recent researches [20, 21, 22] have shown
that both theories of FST and SST can be combined to have
a more flexible and expressive framework for modelling and
processing data and information possessing nonstatistical un-
certainties.

In this paper, by combining the ideas of soft sets and fuzzy
soft sets, the concept of fuzzy soft set-valued maps is initiated;
that is, a map whose range set is a family of fuzzy soft sets.
Thereafter, motivated by the work of Suzuki [29], Djoric and
Lazovic [16], we define the notion of Suzuki-type fuzzy soft
(e, K)-weak contraction in the setting of fuzzy soft sets, where
e is an element of the parameter set E and K ≥ 0. As a re-
sult, fuzzy soft fixed point theorem of Suzuki-type is proved
and some consequences are obtained thereafter. In addition, an
application in homotopy result is established to show the us-
ability of our obtained results.

The rest of this paper is organized as follows. In Subsection
1.1, specific concepts of fuzzy sets and multivalued mappings
are recalled. Subsection 1.2 briefly reviews some background
of soft sets and fuzzy soft sets. Section 2 contains the novel
concepts of fuzzy soft set-valued maps. In Subsection 2.1, a

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M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 28

few fuzzy soft fixed point theorems of Suzuki-type (e, K)-weak
contractions are established. Subsection 2.2 gives some conse-
quences of Subsection 2.1. An application in homotopy result
is presented in Subsection 2.3.

1.1. Fuzzy Sets and Multivalued Mappings

Let (X, d) be a metric space. We denote by C B(X) the class
of all nonempty, closed and bounded subsets of X. Let H(., .)
be the Hausdorff metric on C B(X) induced by d, that is,

H(A, B) = max
{

sup
a∈A

d(a, B), sup
b∈B

d(A, b)
}
,

for A, B ∈ C B(X), where d(x, A) = inf{d(x, a) : a ∈ A}. A point
u in X is a fixed point of a multi-valued mapping T : X −→
C B(X) if u ∈ T u.

Let X be an initial universe. Recall that an ordinary subset
A of X is determined by its characteristic function χA, defined
by χA : A −→{0, 1}:

χA(x) =

1, if x ∈ A0, if x < A.
The value χA(x) specifies whether an element belongs to

A or not. This idea is used to define fuzzy sets by allowing
an element x ∈ A to assume any possible value in the interval
[0, 1]. Thus, a fuzzy set A in X is a set of ordered pair given as

A = {(x,µA(x)) : x ∈ X},

where µA : X −→ [0, 1] = I and µA(x) is called the membership
function of x or the degree to which x ∈ X belongs to the fuzzy
set A. If A is a fuzzy set in X, the (crisp) set of elements in X
belonging to A at least of degree α ∈ (0, 1] is called the α-level
set, denoted by [A]α. That is,

[A]α = {x ∈ X : µA(x) ≥ α}.

On the other hand,

[A]∗α = {x ∈ X : µA(x) > α},

is called the strong α-level set or strong α-level cut. Denote by
I X , the family of all fuzzy sets in X. A mapping T : X −→ I X

is called fuzzy mapping. An element u in X is said to be a
fuzzy fixed point of a fuzzy mapping T if there exists an α ∈
(0, 1] such that u ∈ [T u]α. If there exists α ∈ [0, 1] such that
[A]α, [B]α ∈ C B(X), then

d∞(A, B) = sup
α

H ([A]α, [B]α) .

1.2. Soft Sets and Fuzzy Soft Sets

Let E be the parameter set, A ⊆ E and P(X) represents the
power set of an initial universe of discourse X. In this section,
some basic concepts and examples of soft sets and fuzzy soft
sets are recalled.

Molodstov [26] established the concept of soft sets with the
following definition.

Definition 1.5. [26] A pair (F, A) is called a soft set over X
under E, where A ⊆ E and F is a mapping given by F : A −→
P(X).

In other words, a soft set over X is a parameterized family
of subsets of X. For each e ∈ E, F(e) is considered as the set of
e-approximate elements of (F, A).

Example 1.6. [25] Suppose the following:
X- is the universal set of all students in a certain university,
E- is the set of parameters, given as:

E = {intelligent, hardworking, dull, hardworking and intelligent}.

Assume that they are one hundred students in the university X
given as

X = {x1, x2, x3, x4, x5 · · · x100}, and E = {e1, e2, e3, e4},

where

e1= intelligent, e2 = hardworking,

e3= dull, e4= hardworking and intelligent.

Then F : E −→ P(X) defined by F(e1) = {x1, x2, · · · x10} means
that x1, x2, · · · x10 are intelligent , F(e2) = {x11, x12, · · · x30} means
that x11, x12, · · · x30 are hardworking, F(e3) = ∅ means that
there is no dull student in the university in question, F(e4) =
{x15, x81} means that the students x15 and x81 are both intelli-
gent and hardworking. Then we can view the soft set (F, E)
describing the “kind of students” as the following approxima-
tions:

(F, E) =
{

(intelligent students, {x1, x2, · · · x10}),

(hardworking studnts, {x11, x12, · · · x30})

(dull,∅), (intelligent and hardworking students, {x15, x81})
}
.

Many researchers carried out formal studies of these basic
ideas of soft sets and related notions . For example, Maji etal
[24] developed these notions and established other concepts
such as soft subsets and supersets, intersections and unions,
soft equalities and so on. For soft set-based decision making
approach, the interested reader may consult Cagman and En-
ginoglu [15], Feng and Zhou[17] and Maji et al [23].

Remark 1.7. (See [26]) Every fuzzy set is a special kind of
soft set. In other words, let A be a fuzzy set with membership
function µA. Consider the family of α-level sets given by

Ω(α) = {x ∈ X : µA(x) ≥ α}, α ∈ [0, 1].

If the family Ω is known, then we can find µA(x) via the formula:

µA(x) = sup{α : α ∈ [0, 1]}, x ∈ Ω(α).

This shows that in particular, every fuzzy set is a soft set (Ω, [0, 1]).

In order to model more general scenarios, Maji et al [21]
defined the notion of fuzzy soft sets in the following manner.

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M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 29

Definition 1.8. [21] A pair (F, A) is a fuzzy soft set over X when
A ⊆ E and F : A −→ I X .

Clearly, every soft set can be thought of as a fuzzy soft set.
Following Example 1.6, fuzzy soft sets allow the investigation
of some more intriguing properties such as “how much time
each student works” in which case partial memberships are in-
dispensable.

Example 1.9. [25] Consider Example 1.6. The fuzzy soft set
(F, E) describing the “kind of students” under fuzzy circum-
stances may be given as

(i) F(e1) = {x1/0.5, x2/0.1, x4/0.7},
F(e2) = {x3/0.6, x9/0.7, x20/0.1}

(ii) F(e3) = {x19/0.8, x25/0.1, x4/0.7},
F(e4) = {x13/0.4, x91/0.3, x94/0.1, x97/0.3}.

In what follows, we initiate the concept of fuzzy soft set-
valued maps.

2. Main Results

Let X be an initial universe of discourse, E the parameter
set, I = [0, 1] and I(E;X) denotes the family of fuzzy soft sets
over X under E. By A ∈ I(E;X), we mean the mapping A : E −→
I X , where X is the set of all fuzzy sets in X. Let N, Z, and R
denote the set of nonnegative integers, set of integers and the
set of real numbers, respectively.

Definition 2.1. A mapping T : X −→ I(E;X) is called fuzzy soft
set-valued map.

Notice that if T : X −→ I(E;X) is a fuzzy soft set-valued
map, then for x ∈ X, T x ∈ I(E;X) implies that T x : E −→ I X

is a fuzzy soft set. This further implies that for any e ∈ E,
T (x)(e) : X −→ [0, 1] is a fuzzy set. Hence, (T x)(e)(t) is the
degree of membership of t in (T x)(e). In the remaining part
of this paper, we shall write T [x; e] and T [x; e](t) to represent
(T x)(e) and (T x)(e)(t), respectively.

Example 2.2. Let X = [−4, 4] and E = [0, 5]. Define T : X −→
I(E;X) by

T [x; e](t) =
t2

x2 + e + t2
.

Then T is a fuzzy soft set-valued map. Notice that T [x; e](t) ∈
[0, 1] for all x, t ∈ X and e ∈ E .

Definition 2.3. Let T : X −→ I(E;X) be a soft set-valued map.
Then the e-level set of T is defined by

[T x]eE =
⋃
e∈E

[T [x; e]]eE ,

where

[T [x; e]]eE = {t ∈ X : T [x; e](t) ≥ e}, e ∈ E.

For simplicity, we shall also denote the e-level set of T by [T ]eE ,
whenever the domain of T is obvious.

Proposition 2.4. Every α-level set is an e-level set.

Proof. Let A be a fuzzy set in X and E = (0, 1] be a set of
parameters. Consider a fuzzy soft set-valued map T : X −→
I(E;X) defined by

T [x; e](t) = A(x), for all x, t ∈ X and e ∈ E.

Then for α ∈ (0, 1] = E, we have⋃
α∈(0,1]

[A]α =
⋃

α∈(0,1]

{t ∈ X : A(x) ≥ α}

=
⋃

α∈(0,1]

{t ∈ X : T [x; e](t) ≥ α = e}

=
⋃
e∈E

{t ∈ X : T [x; e](t) ≥ e}

= [T ]eE .

Remark 2.5. Every fuzzy mapping Υ : X −→ I X is a fuzzy soft
set-valued map TΥ : X −→ I(E;X), defined by

TΥ[x; e](t) = {t ∈ X : (Υx)(t) ≥ α}, α ∈ (0, 1].

Notice that I X 7−→ I(E;X) is embedding by Λ −→ ΘΛ, for every
Λ in I X ; where

ΘΛ[x; e](t) = {t ∈ X : Λ(t) ≥ e}.

Let X be a metric space. For x ∈ X, consider two fuzzy soft
sets A, B ∈ I(E;X) and e ∈ E. Assume that [A]eE, [B]

e
E ∈ C B(X).

Then define

PeE (x, A) = inf
{
d(x, a) : a ∈ [A]eE

}
;

PeE (A, B) = inf
{
d(a, b) : a ∈ [A]eE, b ∈ [B]

e
E
}
;

P(A, B) = sup
e

PeE (A, B);

DeE (A, B) = d
H
E ([A]

e
E, [B]

e
E );

where

dHE ([A]
e
E, [B]

e
E ) = max

 supa∈[A]eE d(a, [B]eE ), supb∈[B]eE d(b, [A]eE )
 .

Define a distance function d∞E : I
(E;X) × I(E;X) −→ R by

d∞E ([A]
e
E, [B]

e
E ) = sup d

H
E ([A]

e
E, [B]

e
E ).

Let g : X −→ X be a single-valued mapping and T : X −→
I(E;X) be a fuzzy soft set-valued map. An element x ∈ X is said
to be:

(i) A fixed point of g if x = gx. Denote by Fix(g), the set of
all fixed points of g.

(ii) Fuzzy soft fixed point of T if x ∈ [T x]eE for some e ∈ E.

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M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 30

For x, y ∈ X, define∐
(x, y) = max

{
d(x, y), d(x, [T x]eE ), d(y, [T y]

e
E ),

d(x, [T y]eE ) + d(y, [T x]
e
E )

2
,

d(x, [T x]eE )d(y, [T y]
e
E )

1 + d(x, y)

}
g∐

(x, y) = max
{

d(x, y), d(x, gx), d(y, gy),

d(x, gy) + d(y, gx)
2

,
d(x, gx)d(y, gy)

1 + d(x, y)

}
.∏

(x, y) = min
{
d(x, [T x]eE ), d(y, [T x]

e
E ),

d(x, [T x]eE )d(y, [T y]
e
E )

1 + d(x, y)
.

g∏
(x, y) = min

{
d(x, gx), d(y, gx),

d(x, gx)d(y, gy)
1 + d(x, y)

}
.

Throughout this paper, for E = (0,σ), σ ≥ 1, the function
r : E −→

(
1
2 , 1

)
is defined by

r(e) =

1, if 0 < e < 121 − e, if 12 ≤ e < 1σ.
Motivated by definitions 1.2 and 3 as well as Theorem 1.4, we
give the next Definition.

Definition 2.6. A fuzzy soft set-valued map T : X −→ I(E;X) is
called Suzuki-type soft (e, K)-weak contraction if for all x, y ∈ X
with x , y, there exist some e ∈ E and K ≥ 0 such that

r(e)d(x, [T x]eE ) ≤ d(x, y)

implies

dHE ([T x]
e
E, [T y]

e
E ) ≤ e

∐
(x, y) + K

∏
(x, y).

2.1. Fuzzy Soft Fixed Points of Suzuki-type Soft (e, K)-Weak
Contractions

In this subsection, we first present fuzzy soft fixed theorem
of Suzuki-type fuzzy soft set-valued (e, K)-weak contractions
and then obtain some associated consequences.

Theorem 2.7. Let (X, d) be a complete metric space and T :
X −→ I(E;X) a Suzuki-type fuzzy soft set-valued (e, K)-weak
contraction. If for some x ∈ X, there exists e ∈ E such that
[T x]eE is a nonempty closed and bounded subset of X, then
Fix(T ) , ∅.

Proof. Let e1 ∈ E be such that 0 < e ≤ e1 <
1
σ

, x1 ∈ X and
ρ = 1√

e
. Then by hypothesis, [T x1]eE is nonempty. Therefore,

we can find x2 ∈ X such that x2 ∈ [T x1]eE . Since ρ > 1, choose
x3 ∈ [T x2]eE such that

d(x2, x3) ≤ ρd
H
E ([T x1]

e
E, [T x2]

e
E ).

If x1 = x2, then x1 ∈ [T x1]eE for some e ∈ E, and the theorem is
proved. Assume that x1 , x2 . Since r(e) ≤ 1, therefore,

r(e)d(x1, [T x1]
e
E ) ≤ d(x1, [T x1]

e
E ) ≤ d(x1, x2)

implies

(5)

If d(x1, x2) ≤ d(x2, x3), then from (5), we have

d(x2, x3) ≤
√

ed(x2, x3) < d(x2, x3),

a contradiction. Hence, d(x1, x2) > d(x2, x3) and (5) becomes

d(x2, x3) ≤
√

ed(x1, x2) ≤
√

e1d(x1, x2).

Continuing in this fashion, we generate a sequence {xn}n∈N in X
such that xn+1 ∈ [T xn]eE and

d(xn, xn+1) ≤
√

e1d(xn−1, xn),

from which we have
∞∑

n=1

d(xn, xn+1) ≤
∞∑

n=1

(√
e1

)n−1
d(x1, x2) < ∞.

By a standard argument, we conclude that {xn}n∈N is a Cauchy
sequence in X. The completeness of X implies that there exists
u ∈ X such that xn −→ u as n −→∞.

Claim: for all u , z,we have

d(u, [T z]eE ) ≤ e max
{
d(u, z), d(z, [T z]eE )

}
. (6)

Since xn −→ u as n −→ ∞, there exists a positive integer m
such that

d(xn, u) ≤
1
3

d(u, z), for all n ≥ m.

Given that xn+1 ∈ [T xn]eE , we get

r(e)d(xn, [T xn]
e
E ) ≤ d(xn, [T xn]

e
E )

≤ d(xn, xn+1)
≤ d(xn, u) + d(u, xn+1)

≤
2
3

d(u, z).

Thus, for n ≥ m, we have

(7)

Hence,

r(e)d(xn, [T xn]
e
E ) ≤ d(xn, z) (8)

implies

(9)

From (9), we have

(10)

As n −→∞ in (10), we obtain

d(u, [T z]eE ) ≤e max
{

d(u, z), d(z, [T z]eE ),
d(u, [T z]eE ) + d(u, z)

2

}
+ K min

{
0, d(u, z)

}
.

≤e max
{

d(u, z), d(z, [T z]eE )
}
.

30



M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 31

Now, to show that u ∈ [T u]eE for some e ∈ E, we consider the
following two possible cases:

Case(i): 0 < e < 12 .
Suppose that for all e ∈ E, u , p, u < [T u]eE and p ∈ [T u]

e
E

such that d( p, u) < d(u, [T u]eE ). Setting z = p in (6), we have

d(u, [T p]eE ) ≤ e max
{
d(u, p), d( p, [T p]eE ).

}
(11)

Now,

r(e)d(u, [T u]eE ) ≤ d(u, [T u]
e
E ) ≤ d(u, p)

implies

(12)

Suppose that d(u, p) ≤ d( p, [T p]eE ), then from (12), we have

d( p, [T p]eE ) ≤ ed( p, [T p]
e
E ) < d(p, [T p]

e
E ),

a contradiction. Hence, d(u, p) > d( p, [T p]eE ), and

d( p, [T p]eE ) ≤ ed(u, p) < d(u, p). (13)

Therefore, (11) becomes d(u, [T p]eE ) ≤ ed(u, p). Consequently,

d(u, [T u]eE ) ≤ d(u, [T p]
e
E ) + d

H
E ([T p]

e
E, [T u]

e
E )

≤ d(u, [T p]eE ) + e max
{

d(u, p), d( p, [T p]eE )
}

≤ ed(u, p) + ed(u, p) = 2ed(u, p)
< d(u, p) < d(u, [T u]eE ),(

in view of assumption
)

yields a contradiction. Therefore, u ∈ [T u]eE for some e ∈ E.

Case(ii): 12 ≤ e <
1
σ

.
For this, we shall prove that

(14)

holds for allz ∈ X with z , u. Now, for all m ∈ N , there exists
vm ∈ [T z]eE such that

d(u, vm) ≤ d(u, [T z]
e
E ) +

1
5m

d(u, z).

Thus, we have

d(z, [T z]eE ) ≤ d(z, vm)
≤ d(z, u) + d(u, vm)

≤ d(z, u) + d(u, [T z]eE ) +
1

5m
d(z, u).

Using (6), we have

(15)

If d(z, u) > d(z, [T z]eE ), then from (15), we get

d(z, [T z]eE ) ≤ d(u, z) + ed(u, z) +
1

5m
d(u, z)

=

[
(1 + e) +

1
5m

]
d(u, z),

from which we have[
1

1 + e

]
d(z, [T z]eE ) ≤

[
1 +

1
5(1 + e)m

]
d(u, z).

Using r(e) = 1 − e, we get

(16)

As m −→∞ in (16), we have

r(e)d(z, [T z]eE ) ≤ d(u, z).

On the other hand, if d(u, z) < d(z, [T z]eE ), then (15) gives

d(z, [T z]eE ) ≤ d(u, z) + ed(z, [T z]
e
E ) +

1
5m

d(u, z),

which yields

(1 − e)d(z, [T z]eE ) ≤
(
1 +

1
5m

)
d(u, z). (17)

As m −→∞ in (17), we have

(1 − e)d(z, [T z]eE ) ≤ d(u, z).

This shows that r(e)d(z, [T z]eE ) ≤ d(u, z), which, by Definition
2.6, implies (14). Moreover, since xn+1 , xn for all n ∈ N, then
u , xn+1 . Therefore, setting xn = z in (14), we have

(18)

As n −→∞ in (18), we have

(19)

Since 1 − e > 0, therefore, (19) implies that d(u, [T u]eE ) = 0,
and consequently, u ∈ [T u]eE , for some e ∈ E.

Example 2.8. Let X = [−40, 60], E = (0, 30], e ∈ E be arbi-
trary and d : X × X −→ [0,∞) be defined by

d(x, y) = |x − y|, for all x, y ∈ X.

Define the mapping T : X −→ I(E;X) by

T [x; e](t) =


∆1, if 0 ≤ t ≤

x
30

∆2, if
x

30 < t ≤
x

10

∆3, if
x

10 < t ≤ 60,

where

∆1 =


0.2, if 0 < e1 ≤ 10
0.5, if 10 < e2 ≤ 20
0.8, if 20 < e3 ≤ 30.

∆2 =


0.7, if 0 < e1 ≤ 10
0.9, if 10 < e2 ≤ 20
0, if 20 < e3 ≤ 30.

∆3 =


0.4, if 0 < e1 ≤ 10
0.6, if 10 < e2 ≤ 20
0.2, if 20 < e3 ≤ 30.

31



M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 32

If e = 0.8, then

[T [x; e1]]
0.8
E = {t ∈ X : T [x; e1](t) ≥ 0.8} = ∅.

[T [x; e2]]
0.8
E = {t ∈ X : T [x; e2](t) ≥ 0.8}

=

( x
30
,

x
10

]
.

[T [x; e3]]
0.8
E = {t ∈ X : T [x; e3](t) ≥ 0.8}

=

[
0,

x
30

]
.

Therefore,

[T x]eE =
⋃
e∈E

[T [x; e]]eE

=

[
0,

x
10

]
.

Note that T is a Suzuki-type fuzzy soft set-valued (e, K)-weak
contraction with e = 0.8 and K = 0. In particular, for x = 40
and y = 50,

r(e)d(x, [T x]eE ) = r(0.8)d(40, [T 40]
0.8
E )

= (0.2)d(40, [0, 4]) = 7.2
≤ d(40, 50) = 10

implies

dHE ([T 40]
0.8
E , [T 50]

0.8
E ) = d

H
E ([0, 4], [0, 5]) = 1

≤ 118 = (0.8) max
{

d(40, 50), d(40, [T 40]0.8E ), d(50, [T 50]
0.8
E ),

d(40, [T 40]0.8E ) + d(50, [T 40]
0.8
E )

2
,

d(40, [T 40]0.8E )d(50, [T 50]
0.8
E )

1 + d(40, 50)

}
.

Consequently, Theorem 2.7 can be applied to find 0 ∈ X such
that 0 ∈ [T 0]0.8E .

Corollary 2.9. Let (X, d) be a complete metric space and T :
X −→ I(E;X) a fuzzy soft set-valued map. Assume that for x, y ∈
X with x , y, there exists e ∈ E such that [T x]eE is a nonempty
closed and bounded subsets of X. If

r(e)d(x, [T x]eE ) ≤ d(x, y)

implies

d∞E ([T x]
e
E, [T y]

e
E ) ≤ e

∐
(x, y) + K

∏
(x, y),

then Fix(T ) , ∅.

Proof. Since dHE ([T x]
e
E, [T y]

e
E ) ≤ d

∞
E ([T x]

e
E, [T y]

e
E ), then by

Theorem 2.7, the conclusion holds.

Corollary 2.10. Let (X, d) be a complete metric space and T :
X −→ I(E;X) a fuzzy soft set-valued map. Assume that for x, y ∈
X with x , y, there exists e ∈ E such that [T x]eE is a nonempty
closed and bounded subsets of X. If

r(e)d(x, [T x]eE ) ≤ d(x, y)

implies

dHE ([T x]
e
E, [T y]

e
E ) ≤ e

∐
(x, y),

Then Fix(T ) , ∅.

Corollary 2.11. Let (X, d) be a complete metric space and T :
X −→ I(E;X) a soft set-valued map. If there exist positive con-
stants a, b, c with e = a + b + c < 1, K ≥ 0 such that for x, y ∈ X
with x , y, [T x]eE is a nonempty closed and bounded subsets of
X and

r(e)d(x, [T x]eE ) ≤ d(x, y)

implies

dHE ([T x]
e
E, [T y]

e
E ) ≤ ad(x, y)+bd(x, [T x]

e
E )+cd(d(y, [T y]

e
E ))+K

∏
(x, y).

Then Fix(T ) , ∅.

Corollary 2.12. Let (X, d) be a complete metric space and g :
X −→ X a single-valued mapping. Assume that there exists
0 < e < 1 and K ≥ 0 such that for x, y ∈ X with x , y, g(X) is a
nonempty closed and bounded subset of X . If

r(e)d(x, gx) ≤ d(x, y)

implies

d(gx, gy) ≤ e
g∐

(x, y) + K
g∏

(x, y),

then g has a unique fixed point.

Proof. The existence of the fixed point of g follows from The-
orem 2.7. For uniqueness, suppose that there exist u1, u1 ∈ X
with u1 , u2 such that gu1 = u1 and gu2 = u2. Then

r(e)d(u1, gu1) ≤ d(u1, gu1) = d(u1, u1) = 0 ≤ d(u1, u2)

implies

d(u1, u2) = d(gu1, gu2)

≤ e max
{

d(u1, u2), d(u1, gu1), d(u2, gu2),

d(u1, gu2) + d(u2, gu1)
2

,
d(u1, gu1)d(d(u2, gu2))

1 + d(u1, u2)

}
+K min

{
d(u1, gu1), d(u2, gu1),

d(u1, gu1)d(u2, gu2)
1 + d(u1, u2)

}
≤ e max

{
d(u1, u2), d(u1, u1), d(u2, u2),

d(u1, u2) + d(u1, u2)
2

,
d(u1, u1)d(u2, u2)

1 + d(u1, u2)

}
+K min

{
d(u1, u1), d(u1, u2),

d(u1, u1)d(u2, u2)
1 + d(u1, u2)

}
≤ e max{d(u1, u2), 0} + K(0)
≤ ed(u1, u2),

a contradiction. Therefore u1 = u2.

Next, we present a local fuzzy soft fixed point theorem for
Suzuki-type soft (e, K)-weak contractions. First, recall that an
open ball with radius r > 0 , centered at x0 in a metric space X,
is given by

Br (x0) = {x ∈ X : d(x, x0) < r}.
32



M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 33

Theorem 2.13. Let (X, d) be a complete metric space,
T : Br (x0) −→ I(E;X) be a Suzuki-type fuzzy soft set-valued
(e, K)-weak contraction. Assume that for some x ∈ X, there
exists e ∈ E such that [T x]eE is a nonempty closed and bounded
subset of X and

d(x0, [T x0]
e
E ) < (1 − e)r.

Then Fix(T ) , ∅.

Proof. Let 0 < η < r be such that 0 < (1 − e)(1 +
√

e) ≤
1

1+η , B
∗
η(x0) ⊂ Br (x0) and d(x0, [T x0]

e
E ) < (1 − e)η. There-

fore, (1 − e)η − d(x0, [T x0]eE ) > 0. Choose γ = (1 − e)η −
d(x0, [T x0]eE ) > 0; then there exists x1 ∈ [T x0]

e
E such that

d(x0, x1) < d(x0, [T x0]eE ) + γ. Thus, d(x0, x1) < (1 − e)η. Now,
for ρ = 1√

e
and x1 ∈ [T x0]eE , there exists x2 ∈ [T x1]

e
E such that

d(x1, x2) ≤ ρdHE ([T x0]
e
E, [T x1]

e
E ). Noting that

r(e)d(x0, [T x0]
e
E ) ≤ r(e)d(x0, x1) ≤ d(x0, x1), we have

(20)

If d(x0, x1) ≤ d(x1, x2), then (20) gives

d(x1, x2) ≤
√

ed(x1, x2) < d(x1, x2),

a contradiction. Hence, d(x0, x1) > d(x1, x2) and

d(x1, x2) <
√

ed(x0, x1) <
√

e(1 − e)η.

Notice that by assumption and the triangle inequality, we have

d(x0, x2) ≤ d(x0, x1) + d(x1, x2)
< (1 − e)η +

√
e(1 − e)η

≤
η

1 + η
< η.

Hence, x2 ∈ Bη(x0). Continuing this process recursively, we
generate a sequence {xn}n∈N such that

(a) xn ∈ Bη(x0) for all n ∈ N;

(b) xn ∈ [T xn]eE for each n ∈ N;

(c) d(xn, xn+1) ≤ (
√

e)n(1 − e)η for all n ∈ N.

From (c), it follows by usual arguments that {xn}n∈N is a Cauchy
sequence and converges to some u ∈ Br (x0). From here, fol-
lowing the steps in the proof of theorem 2.7, we conclude that
Fix(T ) , ∅.

2.2. Consequences

Here, we show that there is a link between multi-valued
mappings and fuzzy soft set-valued maps by obtaining some
existing results as consequences of our result.

Corollary 2.14. (See [16, Theorem 2.1]) Let (X, d) be a com-
plete metric space and T : X −→ C B(X) be a multi-valued
mapping. Assume that there exists an α ∈ [0, 1) such that for
all x, y ∈ X,

r(α)d(x, T x) ≤ d(x, y)

implies

H(T x, T y) ≤ αd(x, y);

where the mapping r : [0, 1) −→ (0, 1] is defined by

r(α) =

1, if 0 ≤ α ≤ 11 −α, if 12 ≤ α < 1.
Then there exists u ∈ X such that u ∈ T u.

Proof. Let E := (0, 1) be a parameter set and e ∈ E be arbitrary.
Consider a mapping $ : X −→ E and a fuzzy soft set-valued
map z : X −→ I(E;X) defined by

z[x; e](t) =
$(x), if t ∈ T x0, if t < T x.

Then

[zx]eE =
⋃
e∈E

{t ∈ X : z[x; e](t) ≥ $(x) = e}

= T x.

Therefore,

d(x, y) ≥ r(α)d(x, T x)
= r(α)d(x, [zx]eE ).

Thus, Theorem 2.7 can be applied to find u ∈ X such that u ∈
T u.

From Theorem 2.7 and using the method of proof of Corol-
lary 2.14, we can also derive the following results.

Corollary 2.15. (See [19]) Let (X, d) be a complete metric
space and T : X −→ C B(X) be a multi-valued mapping. As-
sume that there exists α ∈ [0, 1) such that r(α)d(x, T x) ≤ d(x, y)
implies

H(T x, T y) ≤ α max {d(x, y), d(x, T x), d(y, T y)}

for all x, y ∈ X, where the function r is defined as in Corollary
2.14. Then, there exists u ∈ X such that u ∈ T u.

Corollary 2.16. (See [9]) Let (X, d) be a complete metric space
and T : X −→ C B(X) be a multi-valued mapping. Let the
function r : [0, 1) −→ [0, 1) be as defined in Corollary 2.14.
Assume that there exists a(x, y) ∈ [0, 1) such that

r(α)d(x, T x) ≤ d(x, y) implies

H(T x, T y) ≤ a(x, y)
[
d(x, T y) + d(y, T x)

]
for all x, y ∈ X. Then there exists u ∈ X such that u ∈ T u.

Corollary 2.17. Let (X, d) be a complete metric space and T :
X −→ C B(X) a multi-valued mapping. Assume that there exist
constants α ∈ [0, 1) and K ≥ 0 such that for all x, y in X with
x , y, if

r(α)d(x, T x) ≤ d(x, y)

implies

H(T x, T y) ≤ α
∐

(x, y) + K
∏

(x, y),

where the function r is as defined in Corollary 2.14, Then, there
exists u ∈ X such that u ∈ T u.

33



M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 34

Corollary 2.18. Let (X, d) be a complete metric space and T :
X −→ C B(X) a multi-valued mapping. Assume that there exist
constants α ∈ [0, 1) and K ≥ 0 such that for all x, y in X with
x , y, if

r(α)d(x, T x) ≤ d(x, y)

implies

d∞(T x, T y) ≤ α
∐

(x, y) + K
∏

(x, y),

where the function r is as defined in Corollary 2.14. Then, there
exists u ∈ X such that u ∈ T u.

In Corollary 2.17, if T is taken as a single-valued mapping,
then we have the next result.

Corollary 2.19. Let (X, d) be a complete metric space and T :
X −→ X be a single-valued mapping. Assume that there exist
constants α ∈ [0, 1) and K ≥ 0 such that for all x, y in X with
x , y, if

r(α)d(x, T x) ≤ d(x, y)

implies

d(T x, T y) ≤ α
∐

(x, y) + K
∏

(x, y),

where the function r is as defined in Corollary 2.14, Then, there
exists a unique u ∈ X such that u = T u.

2.3. Application in Homotopy
In this section, we apply Theorem 2.13 to prove a homotopy

result. First, for convenience, we recall the following familiar
definitions.

Definition 2.20. A relation ≤ is a total order on a set U if for
all s, t, u ∈ U, the following conditions hold:

(i) Reflexivity: s ≤ s;

(ii) Antisymmetry: if s ≤ t and t ≤ s, then s = t;

(iii) Transitivity: if s ≤ t and t ≤ u, then s ≤ u;

(iv) Comparability: for every s, t ∈ U, either s ≤ t or t ≤ s.

Recall that if the set U satisfies only the axioms (i) − (iii),
then it is said to be partially ordered. In what follows, we shall
call a totally ordered set a chain.

Lemma 2.21. (Kuratowski-Zorn’s Lemma ) If U is any nonempty
partially ordered set in which every chain has an upper bound,
then U has a maximal element.

Definition 2.22. Let X1 and X2 be any two topological spaces
and π,ω : X1 −→ X2 a continuous functions. A function H :
X1 × [0, 1] −→ X2 such that if x ∈ X1, then H(u, 0) = π(u) and
H(u, 1) = ω(u), is called a homotopy between π and ω.

We shall denote the boundary of a set U by Bd(U).

Theorem 2.23. Let (X, d) be a complete metric space and U be
an open subset of X. If M : U × [0, 1] −→ I(E;X) satisfies the
following conditions:

(hom-1) u < M(u, t) for every u ∈ Bd(U) and t ∈ [0, 1];

(hom-2) M(., t) : U −→ I(E;X) is a Suzuki-type soft set-valued
(e, K)-weak contraction for all t ∈ [0, 1];

(hom-3) there exist a nondecreasing function f : [0, 1] −→ R such
that

dHE (M(u, t), M(u, s)) ≤ | f (t) − f (s)| for all s, t ∈ [0, 1]

and each u ∈ U; (21)

(hom-4) M : U × [0, 1] −→ I(E;X) is closed and bounded.

Then M(., 0) has a fixed point.

Proof. Assume p is a fixed point of M(., 0). Then by (hom-1),
p ∈ U. Consider the set

∧
given by∧

= {(t, u) ∈ [0, 1] × U : u ∈ M(u, t)} .

Notice that (0, p) ∈
∧

; hence
∧
, ∅. Let e ∈ E := (0, 1) and

define a partial order ≤ on
∧

as follows:

(t, u) ≤ (s, v) if and only if t ≤ s and d(u, v) ≤
2

1 − e
[
f (s) − f (t)

]
.

Suppose Ω is a chain of
∧

and t∗ := sup{t : (t, u) ∈ Ω}. Assume
that {tn, un} is a sequence in Ω such that (tn, un) ≤ (tn+1, un+1)
and tn −→ t∗ as n −→∞. Then for all positive integers m, n(m >
n),

d(um, un) ≤
2

1 − e
| f (tm) − f (tn)| . (22)

As m, n −→ in (22), we get d(um, un) −→ 0. Hence, {un}n∈N is
a Cauchy sequence and converges to some u∗ ∈ X. Since M
is closed and un ∈ M(un, tn), therefore, u∗ ∈ M(u∗, t∗). From
condition (hom − 1), u∗ ∈ U. Thus, (t∗, u∗) ∈

∧
. Since Ω is

a chain, therefore, (t, u) ≤ (t∗, u∗) for all (t, u) ∈ Ω. In other
words, (t∗, u∗) is an upper bound of Ω. Thus, by Kuratowski-
Zorn Lemma’s ,

∧
has a maximal element (t0, u0). Next, we

show that t0 = 1. Suppose on the contrary that t0 < 1. Let
r = 21−e | f (t) − f (t0)| > 0 with t ∈ (t0, 1] such that Br (u0) ⊂ U.
Notice that by (hom-3),

d(u0, M(u0, t)) ≤ d(u0, M(u0, t0)) + d
H
E (M(u0, t0), M(u0, t))

≤ | f (t) − f (t0)| =
(1 − e)r

2
< (1 − e)r.

Therefore, M(., t) : Br (u0) −→ I(E;X) satisfies all the hypotheses
of Theorem 2.13 for every t ∈ [0, 1]. Consequently, there exists
u ∈ Br (u0) such that u ∈ M(u, t), which implies that (t, u) ∈

∧
for all t ∈ [0, 1]. Now,

d(u0, u) ≤ r =
2

1 − e
| f (t) − f (t0)|,

yields (t0, u0) < (t, u), a contradiction to the fact that (t0, u0)
is maximal. Conversely, assume that M(., 1) has a fixed point;
then on similar steps as above, one can show that M(., 0) has a
fixed point.

34



M. S. Shagari / J. Nig. Soc. Phys. Sci. 2 (2020) 26–35 35

Competing Interests

The author declares that there is no competing interests.

Acknowledgments

The author is thankful to the editors and the anonymous
reviewers for their valuable suggestions and comments on the
manuscript.

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